Differential Equations Practice Exam Questions and Correct Answers (Verified Answers) Plus Rationales 2026 Q&A | Instant Download Pdf

Differential Equations Practice Exam
Questions and Correct Answers
(Verified Answers) Plus Rationales 2026
Q&A | Instant Download Pdf

1. A differential equation that contains only the first derivative is classified as:
A. Second-order differential equation
B. Partial differential equation
C. First-order differential equation
D. Algebraic equation
Answer: C. First-order differential equation
Rationale: The order of a differential equation is determined by the highest
derivative present. If the highest derivative is the first derivative, the equation is
first-order.


2. The equation dydx=3x2\frac{dy}{dx}=3x^2dxdy=3x2 is:
A. Second-order nonlinear
B. First-order ordinary differential equation
C. Partial differential equation
D. Third-order equation
Answer: B. First-order ordinary differential equation
Rationale: The equation contains one independent variable, one dependent
variable, and only the first derivative, making it a first-order ordinary differential
equation (ODE).


3. The order of the differential equation y′′+4y′=xy''+4y'=xy′′+4y′=x is:

,A. First
B. Second
C. Third
D. Fourth
Answer: B. Second
Rationale: The highest derivative present is the second derivative y′′y''y′′, so the
equation is second-order.


4. Which equation is linear?
A. y′=y2y'=y^2y′=y2
B. y′+xy=xy'+xy=xy′+xy=x
C. y′=eyy'=e^{y}y′=ey
D. yy′+1=0yy'+1=0yy′+1=0
Answer: B. y′+xy=xy'+xy=xy′+xy=x
Rationale: A linear differential equation has the dependent variable and its
derivatives only to the first power and not multiplied together.


5. Which equation is separable?
A. y′+y=xy'+y=xy′+y=x
B. dydx=xy\frac{dy}{dx}=xydxdy=xy
C. y′′=xy''=xy′′=x
D. y′+sin⁡(y′)=0y'+\sin(y')=0y′+sin(y′)=0
Answer: B. dydx=xy\frac{dy}{dx}=xydxdy=xy
Rationale: The equation can be rewritten as dyy=x dx\frac{dy}{y}=x\,dxydy=xdx,
allowing separation of variables.

,6. The general solution of dydx=0\frac{dy}{dx}=0dxdy=0 is:
A. y=xy=xy=x
B. y=exy=e^xy=ex
C. y=Cy=Cy=C
D. y=x2y=x^2y=x2
Answer: C. y=Cy=Cy=C
Rationale: If the derivative is zero everywhere, the function must be constant.


7. The integral of dydx=5\frac{dy}{dx}=5dxdy=5 is:
A. y=5x2+Cy=5x^2+Cy=5x2+C
B. y=5x+Cy=5x+Cy=5x+C
C. y=e5xy=e^{5x}y=e5x
D. y=x5+Cy=\frac{x}{5}+Cy=5x+C
Answer: B. y=5x+Cy=5x+Cy=5x+C
Rationale: Integrating the constant 5 with respect to x gives 5x+C5x+C5x+C.


8. Which initial condition specifies a unique solution?
A. y′′=4yy''=4yy′′=4y
B. y′=xy'=xy′=x
C. y′=x, y(0)=2y'=x,\; y(0)=2y′=x,y(0)=2
D. y=x2y=x^2y=x2
Answer: C. y′=x, y(0)=2y'=x,\; y(0)=2y′=x,y(0)=2
Rationale: An initial value specifies the constant of integration, producing a unique
solution.

, 9. The solution of dydx=2x\frac{dy}{dx}=2xdxdy=2x satisfying
y(0)=1y(0)=1y(0)=1 is:
A. y=x2y=x^2y=x2
B. y=x2+1y=x^2+1y=x2+1
C. y=2x+1y=2x+1y=2x+1
D. y=x+1y=x+1y=x+1
Answer: B. y=x2+1y=x^2+1y=x2+1
Rationale: Integrating gives y=x2+Cy=x^2+Cy=x2+C. Using y(0)=1y(0)=1y(0)=1
gives C=1C=1C=1.


10. A homogeneous first-order equation can often be solved by substituting:
A. y=x2y=x^2y=x2
B. x=etx=e^tx=et
C. y=vxy=vxy=vx
D. y=ln⁡xy=\ln xy=lnx
Answer: C. y=vxy=vxy=vx
Rationale: The substitution y=vxy=vxy=vx transforms many homogeneous
equations into separable equations.


11. Which integrating factor is used for
y′+P(x)y=Q(x)y'+P(x)y=Q(x)y′+P(x)y=Q(x)?
A. e−Q(x)e^{-Q(x)}e−Q(x)
B. e∫P(x) dxe^{\int P(x)\,dx}e∫P(x)dx
C. P(x)Q(x)P(x)Q(x)P(x)Q(x)
D. 1P(x)\frac1{P(x)}P(x)1
Answer: B. e∫P(x) dxe^{\int P(x)\,dx}e∫P(x)dx